Sasakian manifold


In differential geometry, a Sasakian manifold is a contact manifold equipped with a special kind of Riemannian metric, called a Sasakian metric.

Definition

A Sasakian metric is defined using the construction of the Riemannian cone. Given a Riemannian manifold, its Riemannian cone is the product
of with a half-line,
equipped with the cone metric
where is the parameter in.
A manifold equipped with a 1-form
is contact if and only if the 2-form
on its cone is symplectic. A contact Riemannian manifold is Sasakian, if its Riemannian cone with the cone metric is a Kähler manifold with
Kähler form

Examples

As an example consider
where the right hand side is a natural Kähler manifold and read as the cone over the sphere. The contact 1-form on is the form associated to the tangent vector, constructed from the unit-normal vector to the sphere.
Another non-compact example is with coordinates endowed with contact-form
and the Riemannian metric
As a third example consider:
where the right hand side has a natural Kähler structure, and the group acts by reflection at the origin.

History

Sasakian manifolds were introduced in 1960 by the Japanese geometer Shigeo Sasaki. There was not much activity in this field after the mid-1970s, until the advent of String theory. Since then Sasakian manifolds have gained prominence in physics and algebraic geometry, mostly due to a string of papers by Charles P. Boyer and Krzysztof Galicki and their co-authors.

The Reeb vector field

The homothetic vector field on the cone over a Sasakian manifold is defined to be
As the cone is by definition Kähler, there exists a complex structure J. The Reeb vector field on the Sasaskian manifold is defined to be
It is nowhere vanishing. It commutes with all holomorphic Killing vectors on the cone and in particular with all isometries of the Sasakian manifold. If the orbits of the vector field close then the space of orbits is a Kähler orbifold. The Reeb vector field at the Sasakian manifold at unit radius is a unit vector field and tangential to the embedding.

Sasaki–Einstein manifolds

A Sasakian manifold is a manifold whose Riemannian cone is Kähler. If, in addition, this cone is Ricci-flat, is called Sasaki–Einstein; if it is hyperkähler, is called 3-Sasakian. Any 3-Sasakian manifold is both an Einstein manifold and a spin manifold.
If M is positive-scalar-curvature Kahler–Einstein manifold, then, by an observation of Shoshichi Kobayashi, the circle bundle S in its canonical line bundle admits a Sasaki–Einstein metric, in a manner that makes the projection from S to M into a Riemannian submersion. While this Riemannian submersion construction provides a correct local picture of any Sasaki–Einstein manifold,
the global structure of such manifolds can be more complicated. For example, one can more generally
construct Sasaki–Einstein manifolds by starting from a Kahler–Einstein orbifold M. Using this observation, Boyer, Galicki, and János Kollár constructed infinitely many homeotypes of Sasaki-Einstein 5-manifolds.
The same construction shows that the moduli space of Einstein metrics on the 5-sphere has at least several hundred connected components.